EG-05 1
Electrical and Magnetic Model Coupling of Permanent Magnet Machines Based on the Harmonic Analysis
M. Merdzan, S. Jumayev, A. Borisavljevic, K. O. Boynov, J. J. H. Paulides, and E. A. Lomonova
Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven 5612 AZ, The Netherlands
A widely used method for the magnetic field calculation in permanent magnet (PM) machines is the harmonic modeling (HM) method. Despite its many advantages, the application of this method is limited to machines in which armature currents, as a source of the magnetic field, are known. Since most of PM machines are supplied with three phase voltages, any variation of parameters in the armature electric circuit could lead to the uncertainty in knowing the armature currents and therefore limit use of the harmonic modeling method. In this paper an approach for overcoming this limitation is presented which enables the calculation of the magnetic field in PM machines without a priori knowledge of the armature currents.
Index Terms—Harmonic modeling, permanent magnet machines, high-speed machines, rotor eddy currents.
I.INTRODUCTION machines is presented. The method is based on a coupling of ARMONIC modeling (HM) technique is a fast and the harmonic model solution with the equation describing the H precise tool for the magnetic field analysis in electrical armature electric circuit. machines [1]. Different types and configurations of machines can be analyzed using this approach, among them rotating permanent magnet (PM) machines [2]-[3]. II.HARMONICMODELINGOFPERMANENTMAGNET Harmonic analysis is based on Fourier representation of MACHINES sources of the magnetic field which have to be expressed in terms of magnetization and current density [1]. In PM Harmonic modeling approach is based on a direct solution machines magnetization source terms correspond to permanent of Maxwell’s equations. By introducing magnetic vector magnets, while current density source terms correspond to potential, the field behavior is described by a single second armature currents. However, PM machines are usually supplied order partial differential equation which is solved by the by voltage sources and the armature currents are not known in method of separation of variables [7]. The domain in which advance. Therefore, they have to be extracted from the applied magnetic field needs to be solved is divided into regions voltage and parameters of the machine electric circuit, such as with different electromagnetic properties [1] in which the resistance and inductance. governing equation takes different forms (Laplace, Poisson, or The resistance and inductance of the armature circuit cannot Helmholtz equation). Field sources are expressed as infinite be considered constant. Because of the skin and proximity series of harmonics and magnetic vector potential solution is effects occurring in armature conductors [4], the resistance assumed to be in the same form. increases with frequency. The armature inductance, on other Equation (1) is the governing equation in its most general ~ hand, decreases with frequency [5]. This is caused, especially form. Magnetic vector potential is indicated with A [Wb/m], ~ 2 in high-speed PM machines, by eddy currents flowing in the current density with J [A/mm ] and remanent flux ~ conducting parts of the machine which influence (mostly) density with Brem [T], while µ [H/m] and σ [S/m] represent the magnetizing inductance. In the work presented here, the permeability and conductivity in considered regions. The change of the armature winding resistance is not analyzed and magnetic field in PM machines has two independent sources: it is assumed to be known in advance. magnets and armature currents. Field components originating To overcome problems related to uncertainty in knowing from each source are solved separately and the total field is the inductance, the magnetic field and armature currents in obtained by the summation. From the point of view of the voltage-fed PM machines could be obtained in several steps permanent magnet field the machine can be divided in two [6]. The inductance can be calculated separately for a given types of regions: with magnetization (permanent magnet) and frequency and then used to calculate the armature currents without magnetization (all others). From the armature field from the supply voltages. Using the calculated currents the point of view, the machine can be divided into a current armature magnetic field can be solved. This method has several carrying region (armature winding), a non-conducting region steps and is not suitable for use in a design procedure where (air gap) and conducting regions (magnet and retaining sleeve fast tools are necessary. if present). In Table I the form of the governing equation In this paper a method for simultaneous solving of the in every region type is shown. In 2D analysis which is magnetic field and the armature currents in voltage-fed PM considered in this paper, the remanence can have a radial and azimuthal component, while the current density and magnetic Corresponding author: M. Merdzan (e-mail: [email protected]). vector potential have only a component in the axial direction. EG-05 2
∞ X pka −pka jβr Aa(r, ϕr, t) = (cr + dr )e (6) ∂A~ ka=1 −∇2A~ + µσ = µJ~ + ∇ × B~ (1) ∂t rem ∞ X pka −pka jβr Using Fourier decomposition, the spatial distribution of Aa(r, ϕr, t) = (cr + dr + Apa(r))e (7)
ka=1 TABLE I GOVERNING EQUATION IN DIFFERENT MACHINE REGIONS where Ipka and Kpka are the modified Bessel functions of first and second kind of order pka, respectively. Terms τ and βr are given by (8), ωar [rad/s] is the angular frequency in Region Governing equation Region Governing equation rotating regions, given by (9), while Apa is the particular part PERMANENT MAGNET FIELD ARMATURE FIELD of the solution in the winding region. Sign ±, originating in (3) indicates that frequency in the rotor increases for the spatial 2 2 ∂A Magnet A = B rem Conducting A = μσ −∇ ∇× ∇ ∂t harmonics rotating in the direction opposite of the rotor. Terms indicated with c and d are unknown constants obtained from Others 2A =0 Air gap 2A =0 ∇ ∇ the boundary conditions. If any of the regions contains a point Winding 2A = μJ −∇ r = 0, the corresponding constant d equals to zero. If term ωar has value zero (the field component rotates in the synchronism conductors in one phase of the armature winding can be with the rotor), the solution (5) takes form of (6). represented as: 3 √ τ = j 2 ωarµσ ; βr = ωart ± pka(ϕr + θ0) (8) ∞ X n(ϕs) = nˆ cos (ka(pϕs + α)) (2) ωar = ωa ± pkaωm (9) ka=1 The solution for the magnetic vector potential of the per- where nˆ is the peak of the winding distribution which depends manent magnet field can be written in the following way (for on the number of turns and harmonic winding factor, k is the a the magnet and all other regions, respectively): harmonic order, p is the number of pole pairs, ϕs [rad] the ∞ angular coordinate in the stator reference frame, while α takes X three different values (0, −2π/3, 2π/3) in three stator phases. Apm(r, ϕr) = (Ahpm(r) + Appm(r)) cos(pkpmϕr) k =1 If balanced 3-phase currents of angular frequency ωa [rad/s] pm flow through the armature windings, the equivalent source of (10) the armature field (composed of infinite number of rotating ∞ waves) can be expressed in complex harmonic form as: X Apm(r, ϕr) = Ahpm(r) cos(pkpmϕr) (11) ∞ X kpm=1 J = Jeˆ j(ωat±pkaϕs) (3)
ka=1 where kpm is the order of the spatial harmonics in the magnet magnetization and Appm is the particular solution in the In (3) + indicates waves rotating opposite of the rotor direc- magnet region. The term Aapm is the homogeneous part of tion, while − stands for waves rotating in the rotor direction. the solution given by: Term Jˆ depends on the amplitude of the armature currents, the pkpm −pkpm peak of the winding distribution and the winding geometry. Ahpm(r) = ar + br (12) In slotless PM machines J represents the current density a b [A/mm2] imposed in the winding region, while in slotted Terms and are unknown constants determined from the machines J is often approximated by a current sheet [A/mm] boundary conditions. In the region which contains the point r = 0 b distributed over the stator inner surface and it is taken into corresponding constant equals to zero. n r = 0 account through boundary conditions [2]. For a domain having regions including the point there is in total 2n − 1 unknown constants and boundary The angular coordinate in the rotor reference frame (ϕr) [rad] can be expressed using ϕ as: conditions. All equations from the boundary conditions are s combined into a matrix equation for the magnet field and, if
ϕs = ϕr + ωmt + θ0 (4) the armature currents are known, for the armature field. The unknown constants are contained in a vector column X and where ωm [rad/s] is the angular velocity of the rotor and are calculated by solving: θ0 [rad] is the initial rotor position. Using Table I and (4), the general solution for the magnetic vector potential of the E(2n−1)×(2n−1) · X(2n−1)×1 = Y(2n−1)×1 (13) armature field can be written for conducting, air gap and From the solution of the vector potential the flux linkage can winding regions, respectively, as: be calculated as [5]: ∞ X I Z π/p A (r, ϕ , t) = (cI (τr) + dK (τr))ejβr (5) ~ a r pka pka ψ = A~ · dl = pls n(ϕs)A(r = rc, ϕs)dϕs (14) ka=1 coil −π/p EG-05 3
where ls is the axial length and rc is the radius at which the The solving procedure is the extension of previously pre- integration is performed, usually in the middle of the winding sented approach for solving the boundary conditions in the region for slotless machines, or stator inner radius for slotted harmonic model. A similar analysis for induction machines machines. After applying (14) flux linkages of the armature was presented in [8]. field, ψa [Vs], and the magnet, ψpm [Vs], can be written as: If the total number of harmonics included in (18) is ktot, it ∞ is possible to define ktot vector rows: X ˆ j(ωat∓kaα) ψa = ψae (15) ka pka −pka F1×(2n−1) = jωaK[0 0 . . . rc rc ] (21) ka=1 where K is a constant dependent on the machine geometry ∞ pka −pka which multiplies terms crc and drc in the armature flux X ˆ j(kpm(p(ωmt+θ0)+α)) ψpm = ψpme (16) linkage amplitude. Furthermore, it is possible to define ktot kpm=ka=1 ka square matrixes E corresponding to E(2n−1)×(2n−1) from ˆ ˆ (13) for every considered harmonic of the armature field and Flux linkages amplitudes ψa and ψpm contain the unknown ka constants from the vector potential solution. The lower sum- in similar way ktot vector columns X corresponding to mation limit in (16) indicates that out of all harmonics in X(2n−1)×1 from (13). Additionally, by dividing Y(2n−1)×1 the permanent magnet field only those of the same order as from (13) by unknown armature current I, it is possible to ka armature harmonics can be linked by the armature winding. define ktot vector columns Y . Moreover N is defined as the zero square matrix with dimensions 2n − 1 and M is the zero vector column of the length 2n − 1. III.COUPLINGOFTHEHARMONICMODELWITHTHE Using introduced matrixes and vectors it is possible to write ARMATUREELECTRICCIRCUIT the following matrix equation: The proposed method is based on the analysis of the arma- ture circuit voltage equation, where the leakage inductance L σ 1 2 k Z F F ... F tot I U [H] is assumed to be independent on frequency and known. The voltage equilibrium equation of one phase of the armature −Y1 E1 N ... N X1 M winding of any PM machine can be written as: di(t) di(t) dψpm(t) v(t) = Ri(t) + L + L + (17) 2 2 2 m σ −Y NE ... N · X = M (22) dt dt dt where v(t) [V] is the supply voltage, i(t) [A] the armature . . . . . . . ...... . . current of the angular frequency ωa, R [Ω] the armature . . . . . . winding resistance and Lm [H] the magnetizing inductance. By using (15) and (16), equation (17) can be rewritten in the −Yktot NNNEktot Xktot M complex notation as: ∞ Multiplication of the first row in (22) with the vector contain- X ˆ ∓jkaα ka U = Z I + jωa ψae (18) ing the unknown current and vectors X gives equation (18). ka ka=1 Only two constants from every X vector are present in the The impedance and the resultant voltage acting on the arma- armature flux linkage expression. Their position in these vec- ture winding are given by (19) and (20). tors determines in which positions the non-zero terms appear in (21). Two required constants for every harmonic have to be Z = R + jωaLσ (19) solved together with other 2n − 3 unknown constants, which requires extension of the matrix to the form shown in (22). ∞ After solving (22) both the armature currents and the X ˆ j(kpm(pθ0+α)) U = V − jpωm kpmψpme (20) unknown constants are calculated simultaneously. By inserting kpm=ka=1 obtained constants in (5) - (7) the magnetic vector potential where V and I in other two phases are shifted for α. In (18) the of the armature field can be calculated and the field in every magnetizing inductance is included in the flux linkage of the region can be obtained using: armature field, and eddy currents are accounted for through B~ = ∇ × A~ (23) the field solution in conducting regions. Permanent magnet flux linkage does not depend on other terms in (18) and can be obtained by solving the unknown constants using (13). IV. RESULTS To solve the unknown constants in the armature field The method is applied to a high-speed slotless PM machine solution (5) - (7), the armature currents, as a source of the with cross section shown in Fig. 1 and parameters shown in armature magnetic field, need to be known. At the same time, Table II. The considered machine has a cylindrical magnet and to obtain the armature currents from (18) constants from the a retaining sleeve made of stainless steel [3]. The results are field solution (contained in the flux linkage amplitude) need to compared to those from a 2D transient finite element model be known. These opposing requirements impose simultaneous (FEM). In Fig. 2 the armature current of one phase obtained solving of both the armature field and the armature currents. by two methods is shown for a same phase voltage. It can be EG-05 4
seen that a good agreement is achieved in both amplitude and 0.08 HM phase which confirms correctness of the proposed method. 0.06 FEM Additionally, the radial component of the flux density in 0.04
J ] 0.02 T [ 6OHHYH r B 0 30 −0.02
SKDVH8 UVL UVO UZ −0.04 M U30 SKDVH9 ] −0.06 0 π/2 π 3π/2 2π SKDVH: ϕs[rad]
EDFNLURQ Fig. 3. The spatial distribution of the radial flux density in the middle of the winding region for the considered PM machine
Fig. 1. The slotless high-speed PM machine used for verification of the model including the magnetic field solution in the voltage equation of the armature winding both the armature currents and the TABLE II magnetic field are solved simultaneously. Therefore, the mag- PARAMETERSOFTHEUSEDSLOTLESSHIGH-SPEED PM MACHINE netic field is obtained directly from applied voltages, without explicit inductance calculation. This makes the method very PARAMETER SYMBOL VALUE useful as an optimization tool for use in PM machines design. Stator inner radius 4.25 mm rsi The method is applicable to PM machines with arbitrary Winding inner radius rw 3.25 mm windings and magnets topology. It can be used for machines Sleeve outer radius rsl 2.75 mm Magnet radius rm 2.25 mm with multiple rotor magnets and account for field solution in Stack length ls 20 mm separate armature slots. Due to increased number of considered Shift between coils of same phase γ 30◦ regions, the model complexity would increase, which, how- Magnet remanence Brem 1.3 T ever, would not limit the suitability of the presented method. Magnet recoil permeability μrec 1.05 Nominal rotational speed n 80000 rpm ACKNOWLEDGMENT
15 The authors would like to thank Micro Turbine Technology Voltage BV for cooperation on the development of a high speed Current HM 10 Current FEM permanent magnet generator for micro-CHP, as well as KIC InnoEnergy for providing research and development funds. 5 [A]
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